Found problems: 1269
2008 Rioplatense Mathematical Olympiad, Level 3, 2
On a line, there are $n$ closed intervals (none of which is a single point) whose union we denote by $S$. It's known that for every real number $d$, $0<d\le 1$, there are two points in $S$ that are a distance $d$ from each other.
[list](a) Show that the sum of the lengths of the $n$ closed intervals is larger than $\frac{1}{n}$.
(b) Prove that, for each positive integer $n$, the $\frac{1}{n}$ in the statement of part (a) cannot be replaced with a larger number.[/list]
2010 Contests, 2
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
2011 Morocco National Olympiad, 2
Solve in $\mathbb{R}$ the equation :
$(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$ $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$ $ 0$.
1991 USAMO, 2
For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that
\[ \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), \]
where ``$\Sigma$'' denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.
2010 Rioplatense Mathematical Olympiad, Level 3, 3
Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy
\[ f(x+y)=f(x)+f(y) \] for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$.
Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.
1986 China Team Selection Test, 2
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
2003 China Team Selection Test, 3
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence.
1984 IMO Longlists, 31
Let $f_1(x) = x^3+a_1x^2+b_1x+c_1 = 0$ be an equation with three positive roots $\alpha>\beta>\gamma > 0$. From the equation $f_1(x) = 0$, one constructs the equation $f_2(x) = x^3 +a_2x^2 +b_2x+c_2 = x(x+b_1)^2 -(a_1x+c_1)^2 = 0$. Continuing this process, we get equations $f_3,\cdots, f_n$. Prove that
\[\lim_{n\to\infty}\sqrt[2^{n-1}]{-a_n} = \alpha\]
1989 USAMO, 5
Let $u$ and $v$ be real numbers such that
\[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \]
Determine, with proof, which of the two numbers, $u$ or $v$, is larger.
2010 Finnish National High School Mathematics Competition, 3
Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.
2009 Kurschak Competition, 3
Find all functions $f:\mathbb{Z}\to \mathbb{Q}$ with the following properties: if $f(x)<c<f(y)$ for some rational $c$, then $f$ takes on the value of $c$, and
\[f(x)+f(y)+f(z)=f(x)f(y)f(z)\]
whenever $x+y+z=0$.
1969 Canada National Olympiad, 1
If $a_1/b_1=a_2/b_2=a_3/b_3$ and $p_1,p_2,p_3$ are not all zero, show that for all $n\in\mathbb{N}$, \[ \left(\frac{a_1}{b_1}\right)^n = \frac{p_1a_1^n+p_2a_2^n+p_3a_3^n}{p_1b_1^n+p_2b_2^n+p_3b_3^n}. \]
2012 China Team Selection Test, 3
Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial
\[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\]
with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality
\[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]
2010 Czech-Polish-Slovak Match, 1
Find all triples $(a,b,c)$ of positive real numbers satisfying the system of equations
\[ a\sqrt{b}-c \&= a,\qquad b\sqrt{c}-a \&= b,\qquad c\sqrt{a}-b \&= c. \]
2007 Germany Team Selection Test, 1
For a multiple of $ kb$ of $ b$ let $ a \% kb$ be the greatest number such that $ a \% kb \equal{} a \bmod b$ which is smaller than $ kb$ and not greater than $ a$ itself. Let $ n \in \mathbb{Z}^ \plus{} .$ Determine all integer pairs $ (a,b)$ with:
\[ a\%b \plus{} a\%2b \plus{} a\%3b \plus{} \ldots \plus{} a\%nb \equal{} a \plus{} b
\]
2011 India IMO Training Camp, 2
Suppose $a_1,\ldots,a_n$ are non-integral real numbers for $n\geq 2$ such that ${a_1}^k+\ldots+{a_n}^k$ is an integer for all integers $1\leq k\leq n$. Prove that none of $a_1,\ldots,a_n$ is rational.
1990 China National Olympiad, 5
Given a finite set $X$, let $f$ be a rule such that $f$ maps every [i]even-element-subset[/i] $E$ of $X$ (i.e. $E \subseteq X$, $|E|$ is even) into a real number $f(E)$. Suppose that $f$ satisfies the following conditions:
(I) there exists an [i]even-element-subset[/i] $D$ of $X$ such that $f(D)>1990$;
(II) for any two disjoint [i]even-element-subsets [/i]$A,B$ of $X$, equation $f(A\cup B)=f(A)+f(B)-1990$ holds.
Prove that there exist two subsets $P,Q$ of $X$ satisfying:
(1) $P\cap Q=\emptyset$, $P\cup Q=X$;
(2) for any [i]non-even-element-subset [/i]$S$ of $P$ (i.e. $S\subseteq P$, $|S|$ is odd), we have $f(S)>1990$;
(3) for any [i]even-element-subset[/i] $T$ of $Q$, we have $f(T)\le 1990$.
2009 Hungary-Israel Binational, 3
Does there exist a pair $ (f; g)$ of strictly monotonic functions, both from $ \mathbb{N}$ to $ \mathbb{N}$, such that \[ f(g(g(n))) < g(f(n))\] for every $ n \in\mathbb{N}$?
1993 USAMO, 4
Let $\, a,b \,$ be odd positive integers. Define the sequence $\, (f_n ) \,$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for $\, n \,$ sufficiently large and determine the eventual value as a function of $\, a \,$ and $\, b$.
2003 China Team Selection Test, 2
Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$,
\[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.
2004 China Team Selection Test, 1
Given sequence $ \{ c_n \}$ satisfying the conditions that $ c_0\equal{}1$, $ c_1\equal{}0$, $ c_2\equal{}2005$, and $ c_{n\plus{}2}\equal{}\minus{}3c_n \minus{} 4c_{n\minus{}1} \plus{}2008$, ($ n\equal{}1,2,3, \cdots$). Let $ \{ a_n \}$ be another sequence such that $ a_n\equal{}5(c_{n\plus{}1} \minus{} c_n) \cdot (502 \minus{} c_{n\minus{}1} \minus{} c_{n\minus{}2}) \plus{} 4^n \times 2004 \times 501$, ($ n\equal{}2,3, \cdots$).
Is $ a_n$ a perfect square for every $ n > 2$?
2010 India IMO Training Camp, 2
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.
2006 Federal Competition For Advanced Students, Part 1, 4
Given is the function $ f\equal{} \lfloor x^2 \rfloor \plus{} \{ x \}$ for all positive reals $ x$. ( $ \lfloor x \rfloor$ denotes the largest integer less than or equal $ x$ and $ \{ x \} \equal{} x \minus{} \lfloor x \rfloor$.)
Show that there exists an arithmetic sequence of different positive rational numbers, which all have the denominator $ 3$, if they are a reduced fraction, and don’t lie in the range of the function $ f$.
2003 Vietnam Team Selection Test, 3
Let $f(0, 0) = 5^{2003}, f(0, n) = 0$ for every integer $n \neq 0$ and
\[\begin{array}{c}\ f(m, n) = f(m-1, n) - 2 \cdot \Bigg\lfloor \frac{f(m-1, n)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n-1)}{2}\Bigg\rfloor + \Bigg\lfloor\frac{f(m-1, n+1)}{2}\Bigg\rfloor \end{array}\]
for every natural number $m > 0$ and for every integer $n$.
Prove that there exists a positive integer $M$ such that $f(M, n) = 1$ for all integers $n$ such that $|n| \leq \frac{(5^{2003}-1)}{2}$ and $f(M, n) = 0$ for all integers n such that $|n| > \frac{5^{2003}-1}{2}.$
1979 IMO Longlists, 53
An infinite increasing sequence of positive integers $n_j (j = 1, 2, \ldots )$ has the property that for a certain $c$,
\[\frac{1}{N}\sum_{n_j\le N} n_j \le c,\]
for every $N >0$.
Prove that there exist finitely many sequences $m^{(i)}_j (i = 1, 2,\ldots, k)$ such
that
\[\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}\]
and
\[m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).\]